December 14, 1883.] 



SCIENCE. 



757 



when the work can be done mechanicall}' and 

 without licsitation, the time occupied iu a com- 

 plete addition of the example, and the mistakes 

 made in it, be carefully noted ; that this be 

 done several times, with an interval of some 

 days between the trials, and the result of each 

 trial kept separate ; that the time and mistakes 

 by the ordinary figures in the same example, 

 iu several trials, be observed for comparison. 

 Please p.ay particular attention to the diti'or- 

 ence in the kind of work required by the two 

 methods in its bearing on two questions, — 

 which of them would be easier to work I13" for 

 hours together, supposing both equally well 

 learned? and in which of them could a reason- 

 able degree of skill be more readily acquired 

 b\- a beginner? The answer to these questions, 

 if the comparison be a fair one, is as little to 

 be doubted as is their high importance. 



Eight volunteer observers to w-hom this 

 example has already been submitted showetl 

 wide ditlerenee in arithmetical skill. One of 

 them took but a few seconds over two minutes, 

 in the best of six trials, to add by the usual 

 figures, and set down the sum, but one figure 

 in all the six additions being wrong ; another 

 added once in ten minutes fift\'-seven seconds, 

 and once in eleven minutes seven seconds, with 

 half the figures wrong each time. The last- 

 mentioned observer had had very little training 

 in arithmetical work, but perhaps that gave 

 a fairer comparison. In the binary figures she 

 made three additions in between seven and 

 eight minutes, with but one place wrong in the 

 three. With four of the observers the binary 

 notation required nearly double the time. 

 These observers were all well practised in com- 

 putation. Their best record, five minutes 

 eighteen seconds, was made by one whose 

 best record was two minutes forty seconds 

 in ordinarv figures. The author's own best re- 

 sults were two minutes thirty-eight seconds 

 binary, and three minutes twenty-three sec- 

 onds usual. lie thus proved himself inferior 

 to the last observer, as an adder, by a system 

 in whicii both were equally well trained ; but a 

 greater familiaritj' (extending over a few weeks 

 instead of a few hours) with methods in binary 

 addition enabled him to work twice as fast with 

 them. Of the author's nine additions by the 

 usual figures, four were wrong in one figure 

 each ; of his thirty-two additions h\ diflferent 

 forms of binary- notation, five were wrong, one 

 of them in two places. One observer found 

 that he required one minute thirty-three sec- 

 onds to add a single column (average of 

 five tried) by the usual figures, and fifteen sec- 

 onds to count the characters in one (average 



of six tried) by the liinary. Though these ad- 

 ditions were rather slow, the results are inter- 

 esting. Tiiey siiow, making allowance for the 

 greater number of columns (three and a third 

 times as many) rc<iuired b}' the binary plan, a 

 saving of nearly half; but they also illustrate 

 the necessity of practice. This observer suc- 

 ceeded with the binary arithmetic by avoiding 

 the sources of delay that particularly embar- 

 rass the beginner, b}- contenting himself with 

 counting only, and not stopping to divide by 

 two, to set down an unfamiliar character, or to 

 recognize the mark by whicii ho must distin- 

 guish his next column. One well-known mem- 

 ber of the Washington philosophical society 

 and of the American association for the ad- 

 vancement of science, who declined the actual 

 trial as too severe a task, estimated his prob- 

 able time with ordinary figures at twenty min- 

 utes, with strong chances of a wrong result, 

 after all. 



These statistics prove the existence of a 

 class of persons who can do faster and more 

 reliable work b}- the binary reckoning. But 

 too much should not be made of them. Let 

 them serve as specimens of facts of which a 

 great man}- more are to be desired, bearing on 

 a question of grave importance. Is it not 

 worth our while to know, if we can, bj- impar- 

 tial tests, whether the tax imposed on our 

 working brains by the system of arithmetic in 

 dail}- use is the necessary price of a blessing 

 enjoyed, or an oppression? If the strain pro-' 

 duced by greater complexity and intensit3' of 

 mental labor is compensated bj- a correspond- 

 ingly greater rapidity in dealing witii figures, 

 the former may be the case. If, on the con- 

 trary', a little practice suffices to turn the bal- 

 ance of rapidity, for all but a small body of 

 highly drilled experts, in favor of an easier 

 system, the latter must be. This is the ques- 

 tion that the readers of Science are invited to 

 help in deciding. The difficulties attending a 

 complete revolution in the prevalent sj'stem 

 of reckoning are confessedly stupendous ; but 

 they do not render undesirable the knowledge 

 that experiment alone can give, whether or not 

 the cost of that system is unreasonably high ; 

 nor should thev prevent those who accord them 

 the fullest recognition from assisting to furnish 

 the necessary' facts. 



Those who are willing to undertake the ad- 

 dition on the i^lan proposed or on an}- better 

 plan, or who will submit it to such .acquaint- 

 ances, skilled or unskilled, as maj* be persuaded 

 to take the trouble to learn the mechanism of 

 binary- adding, will confer a great favor by in- 

 forming the writer of the time occupied, and 



